Summary
The Helmholtz-Kirchhoff ODEs governing the planar motion ofN point vortices in an ideal, incompressible fluid are extended to the case where the fluid has impurities. In this case the resulting ODEs have an additional inertia-type term, so the point vortices are termed massive. Using an electromagnetic analogy, these equations also determine the behavior of columns of charges in an external magnetic field. Using the symmetries, we reduce the four degrees of freedom system of two “massive” vortices totwo degrees of freedom. We exhibit an integrable case and a nonintegrable one, according to choices of parameters. Nonintegrability is verified using a recent result obtained independently by Lerman and by Mielke, Holmes, and O'Reilly. Finally, we discuss the behavior of solutions as the masses of the vortices tend to zero, using for initial conditions a point of the trajectory of the Helmholtz-Kirchhoff equations.
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Communicated by Jerrold Marsden
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Ragazzo, C.G., Koiller, J. & Oliva, W.M. On the motion of two-dimensional vortices with mass. J Nonlinear Sci 4, 375–418 (1994). https://doi.org/10.1007/BF02430639
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DOI: https://doi.org/10.1007/BF02430639